Mathematical Background and Formulations
2.1 Preliminaries and Basic Concepts
Denote by N the set of all natural numbers, R+ the set of all non-negative real numbers, Rn the n-dimensional Euclidean space and its norm kxk = pPn
i=1x2i for every x ∈ Rn, Rn×m the set of all n × m real matrices. Let C([a, b], D) (PC([a, b], D)) denote the space of continuous (piecewise continuous) functions mapping [a, b], with a < b for any a, b ∈ R+, into D, for some open set D ⊆ Rn. Also, denote by C1,2(R+× Rn; R+) the space of all real-valued functions V (t, x) defined on R+× Rn such that they are continuously differentiable once in t and twice in x.
Consider the following system of time-variant ordinary differential equations (ODE),
˙x(t) = f (t, x(t)), (2.1)
where x ∈ Rn, t ∈ R+, and f : I × D → Rn is continuous on I × D with I ⊆ R+ and D ⊆ Rn such that it contains the origin. For a given (t0, x0) ∈ I × D with x0 = x(t0), the corresponding initial-value problem (IVP) related to equation (2.1) can be written as
˙x(t) = f (t, x(t)),
x(t0) = x0. (2.2)
A continuously differentiable function φ(t) defined on an interval I ⊂ R+such that φ(t) ∈ D for all t ∈ I is said to be a solution of the IVP (2.2) if ˙φ(t) = f (t, φ(t)) for all t ∈ I, and φ(t0) = x0 with t0 ∈ I.
Theorem 2.1.1. If f is continuous on I × D, then for any (t0, x0) ∈ I × D there exists at least one solution of the IVP (2.2) in I.
Definition 2.1.2. A function f (t, x) defined on I × D is said to be locally Lipschitz in x if there exists a constant L > 0, called Lipschitz constant, such that
kf (t, x1) − f (t, x2)k ≤ Lkx1− x2k
for any points (t, x1) ∈ I × D and (t, x2) ∈ I × D. Moreover, if this inequality holds for all x ∈ Rn, then f is said to be globally Lipschitz in x.
Theorem 2.1.3. If f (t, x) is continuous with respect to first variable and locally Lipschitz continuous with respect to second variable, then for any (t0, x0) ∈ I × D there exists a unique solution of the IVP (2.2).
A point xeq∈ Rn is said to be an equilibrium point of the differential equation in (2.1), or trivial solution of system (2.2) if f (t, xeq) = 0 for all t ≥ t0. Since any equilibrium point can be shifted to the origin, throughout this thesis we will deal xeq ≡ 0 (or x ≡ 0, for simplicity of notation). In the following, we state the definitions of some stability concepts.
Definition 2.1.4 (Stability). Let x(t) = x(t, t0, x0) be the solution of IVP (2.2) for any t ≥ t0, then the trivial solution x ≡ 0, is said to be
(i) stable if, for any > 0 and t0 ∈ R+, there exists a δ = δ(t0, ) > 0 such that
kx0k < δ implies kx(t)k < for any t ≥ t0;
(ii) uniformly stable if (i) holds with δ = δ();
(iii) asymptotically stable if (i) holds and there exists a positive constant δ = δ(t0) such that
kx0k < δ implies lim
t→∞x(t) = 0;
(iv) globally asymptotically stable if (iii) holds with an arbitrary large constant δ;
(v) uniformly asymptotically stable if it is uniformly stable and there is a positive constant δ, independent of t0, such that, for all kx0k < δ, limt→∞x(t) → 0, uniformly in t0; that is, for any η > 0, there is T = T (η) > 0 such that, for all kx0k < δ,
kx(t)k < η, ∀t ≥ t0+ T ;
(vi) globally uniformly asymptotically stable if (v) holds with an arbitrary large constant δ;
(vii) exponentially stable if there exist positive constants δ, k and λ such that kx(t)k ≤ kkx0ke−λ(t−t0), whenever kx0k < δ, and t ≥ t0;
(viii) globally exponentially stable if (vii) holds with an arbitrary large constant δ;
(ix) unstable if (i) does not hold.
In the following, we define some important classes of function that will be used in rest of the thesis.
Definition 2.1.5. Let D ⊂ Rn be an open set containing x = 0. A function W : D → R is said to be positive-definite on D if it is continuous on D, W (0) = 0, W (x) > 0 for x ∈ D\{0}; it is said to be radially unbounded if it is positive-definite and W (x) → ∞ as kxk → ∞.
Definition 2.1.6. The upper right-hand Dini derivative of V (t, x) that is continuous in t and locally Lipschitz in x along the solution of (2.1) is defined by
D+V (t, x) = lim
h→0+sup 1
hV (t + h, x + hf (t, x)) − V (t, x).
Furthermore, if V (t, x) has continuous partial derivatives with respect to t and x, This derivative becomes
V (t, x) =˙ ∂V (t, x)
∂t + ∇xV (t, x) · f (t, x)
where ∇xV (t, x) is the gradient vector of V with respect to x.
Definition 2.1.7. A continuously differentiable function V : D → R is said to be a Lyapunov function if it is positive-definite and non increasing in its domain, i.e.,
V (0) = 0, V (x) > 0, for x ∈ D\{0} and ˙V ≤ 0 in D. (2.3)
The following theorem provides sufficient conditions to guarantee stability and asymp-totic stability of the autonomous system
˙x(t) = f (x), (2.4)
Theorem 2.1.8. Let x = 0 be an equilibrium point of (2.4), and V : D → R where D contains x = 0 be a continuously differentiable function satisfying (2.3). Then, x = 0 is stable. It is said to be asymptotically stable if
V < 0˙ in D\{0}.
Consider the positive-definite function V (x) = xTP x, where P is a positive-definite matrix. Then, the following inequalities hold
λmin(P )kxk2 ≤ xTP x ≤ λmax(P )kxk2 (2.5)
where λmin(P ) and λmax(P ) are the maximum and minimum eigenvalues of P , respectively.
Consider the linear system
˙x = Ax, t ≥ 0
x(0) = x0 (2.6)
where x ∈ Rn and A ∈ Rn×n. The stability properties can be analyzed by Lyapunov stability theory as follows. Define V (x) = xTP x as a Lyapunov function candidate of system (2.6), where P is a positive-definite matrix satisfies the Lyapunov equation
ATP + P A = −Q
with Q being a positive-definite matrix. Then, the derivative of V (x) along the trajectories of (2.6) is given by
V = ˙x˙ TP x + xTP ˙x = xT(ATP + P A)x = −xTQx < 0,
which implies that the equilibrium point of system is asymptotic stable.
In practice, transforming physical phenomena into mathematical models often includes uncertain factors due to modelling mismatches, linearization, approximations or measure-ment errors, etc. It has been realized that considering such uncertainties results in more accurate systems; see for instance [16,46,57,97,176]. Consider the uncertain system
˙x = (A + ∆A)x, t ≥ t0
x(t0) = x0, (2.7)
where ∆A is a piecewise continuous function representing parameter uncertainty with bounded norm, we always assume the following assumption holds throughout this thesis Assumption A. The admissible parameter uncertainties are defined by
∆A(t) = DU (t)H, ∀ t ∈ R+,
with D, H being known real matrices with appropriate dimensions that give the structure of the uncertainty, and U (t) being unknown real time-varying matrix representing the uncertain parameter and satisfying kU (t)k ≤ 1.
To further analyze the stability properties of nonlinear systems, the following class functions, also known as comprison functions, are needed [79].
Definition 2.1.9. A function α ∈ C([0, a], R+) is said to be in class K if α(0) = 0, and it is strictly increasing. It is said to be in class K∞ if it is in class K, a = ∞, and α(r) → ∞ as r → ∞.
Definition 2.1.10. A function β ∈ C([0, a] × R+, R+) is said to belong to class KL if β(·, s) ∈ K for each fixed s, β(r, ·) is decreasing for each fixed r, and β(r, s) → 0 as s → ∞.
Assume that x ≡ 0 of the nonlinear system (2.2) is asymptotically stable. If this system undergoes a bounded-energy disturbance input w ∈ PC(R+, Rm), what can be said about the qualitative behaviour of the output x of the forced system
˙x = f (t, x, w), t ≥ t0
x(t0) = x0 (2.8)
where f : [0, ∞) × Rn× Rm → Rn is piecewise continuous in t and locally Lipschitz in x and w. The input w(t) is a piecewise continuous bounded function of t for all t ≥ 0.
The following definition gives an answer to this question.
Definition 2.1.11 (Input-to-State Stability [79]). System (2.8) is said to be ISS if there exist functions β ∈ KL and γ ∈ K such that, for any x0 and w, the solution x(t) exists for all t ≥ t0 and satisfies
kx(t)k ≤ β(kx0k, t − t0) + γ( sup
t0≤s≤t
kw(s)k). (2.9)
Clearly, for a large enough t, β → 0 and the solution will eventually be bounded by a class K function γ, which depends on the input. One can easily notice that if input w(t) = 0, for all t ≥ t0, the ISS property reduces to the classical asymptotic stability of the trivial solution of the corresponding unforced system.
To analyze the ISS of (2.8), one can use Lyapunov-type theorem to provide a set of sufficient conditions as follows:
Theorem 2.1.12. [79] Let x(t) = x(t, t0, x0) be the solution of (2.8). Let V : [0, ∞) × Rn → R be a continuously differentiable function such that the following conditions hold for any (t, x, w) ∈ R+× Rn× Rm
α1(kxk) ≤ V (t, x) ≤ α2(kxk) (2.10)
∂V (t, x)
∂t +∂V (t, x)
∂x f (t, x, u) ≤ −W (x), ∀kxk ≥ ρ(kwk) > 0 (2.11) for all (t, x, w) ∈ [0, ∞) × Rn× Rm where α1, α2 are class K∞ functions, ρ is a class K function, and W (x) is a continuous positive definite function on Rn. Then, system (2.8) is ISS with γ = α−11 ◦ α2◦ ρ.